Image Reconstruction Algorithm for Electrical Capacitance Tomography

INTRODUCTION

The formation and development of ECT (Electrical Capacitance Tomography) technology are in the late 1980s. It has many advantages, such as non-invasive, simple structure, low cost, fast response, good safety performance, wide application and so on. Image reconstruction algorithm is a key technology of ECT. Solving this kind of problems has some difficulties because of the restriction of less independent capacitance measurements, soft market characteristics, the non-linear of the problem to be solved and so on. The distance to the requirements of industrial application is farther. So researching good image reconstruction is important and urgent (Masoudzadeh and Khalilian, 2007).

This study gives a kind of image reconstruction algorithm based on Chebyshev neural networks for electrical capacitance tomography. Experimental results show that the algorithm is effective. In terms of the reconstruction object this study investigated, the quality of this image reconstruction is better than LBP. Landweber and conjugate gradient algorithm. So it provided a new and an effective method for the ECT image reconstruction.

THE BASIC PRINCIPLES OF ECT SYSTEM

In the schematic of a typical 12-electrode ECT system is shown in Fig. 1. There are three basic components in the ECT system, capacitive transducer, capacitance data acquisition system, image reconstruction calculation (Haiqing and Zhirao, 2000).

Fig. 1:	System structure of ECT

The basic principle is that each-phase in the multi-phase medium with different dielectric and the capacitive transducer installed outer wall of the insulated pipeline get the electrode capacitance (Larbi and Abdelkader, 2005). Due to the capacitance reflects the distribution of dielectric constant inside the tubes, the computer can reconstruct the image according to a certain algorithm with the data, get more phase distribution on the intuitive and get more information about the distribution of multi-phase flow.

Analysis the capacitance field sensitivity unitary, in order to reduce the error between the theoretical data and the measured data, the distribution of the normalized capacitance field sensitivity between i and j plate is defined by the following form:

In the formula, ε₂ and ε₁ are the dielectric constant of the two-phase flow inside the tubes (ε₂>ε₁), C_{i, j} (t) is the dielectric constant of unit t, the electrode capacitance between i and j plate is ε when other dielectric constants are ε; the electrode capacitance are C_{i, j} (ε₂), C_{i, j} (ε₁) between i and j plate when the plate are filled with the medium ε₂ and ε₁. μ (t) is the area reciprocal of unit t. Nein is the numbers of the units in the pipeline (Lihui et al., 2004).

The distribution of capacitance field sensitivity represents the change of the electrode capacitance when the area of the unit changed from ε₁ to ε₂. It is the changed in density of normalized capacitance. In this thesis, 66 independent capacitance components are got by finite element method with capacitance-sensitive 12-electrode array (Hayati, 2007), the sensitive field is subdivided in the form of triangulation, as it is shown in the Fig. 2, there are 192 subdivision units.

THE CHEBYSHEV ALGORITHM OF THE ECT SYSTEM

Chebyshev neural network model: MIMO Chebyshev neural network model is shown in Fig. 3. The dimension of the input vector n is 66, that the value of the sensor capacitance is 66. The hidden layer unit number L is the number of support vector, which is done L times inner product. The output vector is corresponded to 192 gray split units in the sensitive field, that is m = 192.

Chebyshev neural network not only has a simple algorithm, a high speed convergence of learning process but also has excellent characteristics in the linear and nonlinear accurate approximation. Therefore, this model has played a large role in promoting the application of neural network. In this paper, a modified Chebyshev neural network model is advanced, which is not only simple, fast learning convergence but also the network input values can be any value (Taiwo and Abubakar, 2011). So it is a MIMO multi-layer feed-forward neural network model and it has excellent characteristics in the linear and nonlinear accurate approximation.

The connection weights from input layer to the hidden layer are constant 1. The weights from hidden layer to output layer are:

In the formula, m is the number of the output neurons, g is the number of the hidden neurons, W_{i, j} is the number of the connection weights from the number i hidden layer to the number j output layer, the input sample is uk = [x1, k, x2, k, ..., xn, k]^T (k = 1, 2, ..., s),

In the formula, n is the number of the input neurons, s is the number of the input sample, the actual output is:

Fig. 2:	12-electrode capacitance sensor subdivision map

Fig. 3:	MIMO Chebyshev neural network model

Desired output is:

Neural network output:

(1)

In the Eq. 1 pi is the number i neuron of the hidden layer (or node), the function is Chebyshev orthogonal polynomials (Navin et al., 2009), that is:

X is the unipolar S function:

but:

The input range [-∞,+∞] can be transformed into the output range [0, 1] by the unipolar S function. The gradient of the S function can be changed through modifying σ. Neuron output error E_{j, k}:

(2)

Network performance objective:

(3)

Number j line weight adjustment:

(4)

In Eq. 4:

η is the learning rate (0<η<1).

Features sample extraction: The basic idea is iterating and adjusting the N samples of sample set U constantly through the improved k-means clustering algorithm and clustering it for M class sample sets Ai (i = 1, 2, ..., M). Sample clustering is shown in Fig. 4.

The distance between the samples points used Euclidean distance:

In the above equation, d (X, X') is the distance between the sample X = [x₁, x₂, ..., x_m] and X' = [x'₁, x'₂, ..., x'_m].

Assuming the sample set U has N samples, it will be gathered for M class, the improved algorithm is described below.

(5)

In the Eq. 5, j is the sample number, is the input data, r is the sample number which is the nearest with.

(6)

In the Eq. 6, β is the learning rate, int is integer operator. Training m samples once, meanwhile, turning down the learning rate.

(7)

Fig. 4:	Sample clustering

Fig. 5:	Training line graph

Learning algorithm described: As in conclusion, the overall design of image reconstruction algorithm is as follows.

All training samples input periodically until the network convergence and the total output error is less than allowable value.

Chebyshev neural network simulation example: In order to proving the validity of the Chebyshev neural network, we give an application example. To consider the nonlinear function’s Chebyshev neural network training model:

We choose the number of training sample is 300, the learning rate is 0.1, the number of the hidden layer is 20. Training line graph which the training effects close with target 0 with the changes of the training sample number is shown in Fig. 5.

IMAGE RECONSTRUCTION SIMULATION RESULTS

In the simulation experiment, the input capacitance of Chebyshev neural network is 66, output is 792 pixels in the imaging area of the pipeline. In order to obtain more accurate results, according to the different distribution in the pipeline, output is divided into six different parts, each part is 132 pixels. Finally, combined the output of the six sub-network. In the simulation, 501 samples are selected as the sample set, including stratified flow, kernel streaming, annular flow, eccentric flow and two samples of objects. The samples in the training set are obtained through simulation.

The simulation object is the two-phase medium distribution, hidden layer weight adjustment function:

The threshold value for each output unit in this neural network is 3.5, higher than 3.5 is regarded as high dielectric. Otherwise, lower than 3.5 is regarded as low dielectric. The quality of image reconstruction is evaluated through accuracy, the percentage proportion of the correct number of pixels in the total number of pixels inside the tubes.

According to the simulation principles above, reconstructing the samples in the training sample set by using Chebyshev neural network and comparing with the simulation results of the FLBP neural network and RBF reconstruction algorithm. It is shown in Fig. 6. In Fig. 6, the black region stands for the high dielectric (dielectric constant is 6) and the white region stands for the low dielectric (dielectric constant is 1).

The phase separation concentration comparison of different algorithm is shown in Table 1 and the space images relative error of different algorithm is shown in Table 2, the average accuracy of the reconstruction algorithm which we select can reach 99%. One view is the accuracy of image reconstruction by using Chebyshev algorithm is higher than FLBP and RBF image reconstruction algorithm.

Fig. 6:	Image reconstruction simulation results

Table 1:	Phase separation concentration comparison of different algorithm

Table 2:	Space images relative error of different algorithm